ALL
RIGHTS
RESER
VED.
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Le
c
t
u
r
e
12
20
23
.
09
2
2
$
4
.
9
Lin
ea
r
Appr
o
ximatio
ns
$
4
.
10
Ta
y
l
o
r
Po
l
y
n
o
m
a
i
l
s
The
ide
a
of
line
ar
appr
o
ximation
is
ve
r
y
simple
:
Fo
r
a
smooth
funct
ion
fix
)
nea
r
x
=
a
,
we
use
it
s
ta
n
g
e
n
t
line
to
appr
o
ximat
e
it
.
xY
Q
fla
+
ox
)
2
yo
y
*
3
wy
-d
y
=
Ex
)
((X)
=
f(
a)
+
f(
(
a)
(
X
-
a)
↳
Y
·
-
dy
=
fix
)
dx
P
fa
l
·
↓
s
x
=
dx
X
-
7
a
a
+
0x
Ho
w
to
writ
e
the
ta
n
g
e
n
t
line
quation
?
·
Pa
s
s
i
n
g
thr
o
ug
h
p
=
(a
,
f(
al
·
wit
h
sl
ope
f'(
a)
=>
Y-
fla)
=
f'(
a)
(
X
-
a)
we
re
d
e
f
i
n
e
such
by
Li
x
s
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Why
is
linear
appr
o
ximation
impor
t
ant
?
·
fix
s
might
be
ve
r
y
non-linear
/complicat
ed
but
Le
x
)
giv
es
essential
be
haviour
of
fix
s
near
x
=
a
.
ef
f
icient
fo
r
comput
ation
If
it
is
no
t
good
enough
,
we
need
higher
or
der
appr
o
ximation
i
Ta
y
l
o
r
polyn
oma
i
l
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
-
Ex
ample
Us
e
the
linearization
fo
r
x
at
x
=
25
to
find
an
appr
o
ximation
va
l
u
e
of
EC
.
d
fix
=
r
fix
)
=
xs
=
The
ta
n
g
e
n
t
line
of
fix
s
at
x
=
25
is
th
e
line
passin
g
tho
ugh
P
=
(25
,
55)
wit
h
slo
pe
=
=
(25)
-=
to
So
(Ix
=
55 +
to
(
X
-
2
5
)
.
Put
t
ing
x
=
26
50
=
f(
26
)
=
((26)
=
55
+
%(26
-
25)
=
5
.
1
.
1]
Remark
The
er
r
or
te
r
m
is
defined
by
Err
or
&
true
va
l
u
e
-
appr
o
ximation
=(x) I
f
f(
x
)
-
L(x)
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Theor
em
11
<Err
o
r
fo
r
m
u
l
a
fo
r
linearization)
su
ppo
se
that
f"
it
s
ex
i
s
t
s
fo
r
an
int
er
v
al
I
and
a
,
x
=
I
.
Then
Is
be
t
w
e
e
n
a
and
x
suc
h
that
E(x1
=
I
(x
-
as
,
wi
t
h
EIx
fix
-L
IX)
,
and
2x
1
=
f(
a)
+
fi(
a)
(
x
-
a)
.
Pr
oof
:
we
ca
n
as
su
me
x
>
a
.
The
ca
r
e
x<
a
is
similar
.
Then
apply
the
Ge
ne
r
al
i
z
e
d
Mean-v
alue
The
or
e
m
E(X
)
EIX
)
-
E(a)
=(U)
=-
-
f
in
s-
f
ia
l
--
-
-
(x
-
a)
X-
a
s
-
(a
-
as
21
4
-a)
214
-
a)
↑
fo
r
some
u
=
(a
,
x)
=
I
-"
(
s
)
an
d
s
=
(a
.
4)
I
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Now
f(
x
)
=
LIx)
+
EI
X
)
=
LI
x
)
+
I
(x -
as
So
If
f"
(
t
)
>0
fo
r
t
be
t
e
e
n
a
and
X
Co
n
v
e
x
S
then
fix
)
>
Lix)
-
If
f"
(
t
)
<
0
fo
r
t
be
t
e
e
n
a
and
x
then
fix
)
<
L(
X
)
conca
v
e
-
Theor
em
If
f(
x
)
<
0
fo
r
all
xe
I
int
er
v
al
then
o
is
co
n
v
e
x
on
1.
~
If
f)
<
0
fo
r
all
x
=
1
,
then
f
is
con
ca
v
e
.
-
Recall
:
I
is
cal
l
ed
conv
e
x/concav
e
on
I
if
F
X
,,
x2
=
2
,
an
d
0 f
t=1
,
we
ha
v
e
f(
t
x
,
+
x
-
t)
x
2)
=
(
,
)
tf
(xx)
+
(1
-
t)f
(x2)
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Ex
ample
·
Le
t
fix
)
=
ex
,
g(x)
=
en(X)
,
hix1
=
x
!
Stu
d
y
their
conv
e
x/concav
e
pr
o
per
ty
.
·
Ho
w
about
wixi=
a
Y
fo
r
some
po
sitiv
e
const
ant
a
?
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
S
Ta
y
l
o
r
Po
l
y
n
o
m
a
i
l
s
The
idea
is
also
simple
:
Fo
r
a
smo
o
th
funct
ion
y
=
fix
)
ne
ar
x
=
a
·
The
dom
inat
e
appr
o
ximation
is
giv
en
by
linear
te
r
m
of
(x
-a
s
mo
r
e
pr
ecisely
flas
+
fa
s
(
x
-
a
s
(
·
If
we
use
higher
or
der
te
r
m
(x
-
as"
we
could
get
a
be
t
t
e
r
appr
o
ximation
·
what
is
the
coefficient
be
f
or
e
(x
-
as"
?
(a)
2
·
Ev
en
higher
or
de
r
te
r
m
?
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
De
finit
ion
If
f(
x
)
ex
i
s
t
s
on
an
op
en
int
er
v
al
cont
aining
x
=
a
,
then
the
po
lyno
mail
Pne
x
s
+
fial
+
1
"
(x -
a)
+
(x
-
as
+
...
+
"(x
-
is
called
th
e
n-
th
or
de
r
Ta
y
l
o
r
po
lyn
o
mal
fo
r
-
nea
r
a.
Pr
op
e
r
t
y
.
Puix
s
mat
ches
fix
,
at
a
up
to
or
de
r
n
.
Mor
e
pr
ecisely
:
Pr(
a)
=
f(
a)
Pn(
a)
=
-(
a)
pas
fa
s
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
And
we
hav
e
f(
x
)
=
PnIX)
.
De
fine
En
1x)
=
fix
)
-
PuIX)
n-t
h
or
de
r
Er
r
o
r
te
r
m
Theor
em
12
If
the
(n
+
1)
-
th
or
de
r
de
riv
at
iv
e
fint
)
,
-)
ex
i
s
t
s
fo
r
+
in
an
int
er
v
al
cont
aining
x
and
a
,
then
(((x
-
ag+
En(x1
=
(n+
1)
!
fo
r
com
e
be
t
w
e
e
n
a
and
X
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Def
(Big
-O
No
t
atio
n)
We
writ
e
f(
x
)
=
0
(u
x)
as
x
-
a
if
/f
ixs/a
Kl
uvs
l
holds
fo
r
X
in
a
neibourhood
of
a
and
so
me
const
ant
12
.
Ex
ample
.
As
x-
O
3
ae
*
=
1
+
x
+
!
+
+
..
+
+
0x)
2
X
xY
(b)
20S
X
=
1
-
I
!
I
!
.
.
+
E1)"
+
Olx2
n
t'
I
x
5
1
sin
x
=
x
-
+
5
!
+
...
+
7)
,
+0
1x
id
x
=
1
+
x
+
x4
...
+
x"
+
0(x
4+)
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
(e)
x
=
I
=
1
+
(
-
x)
+
1
-
x4
-
..
+
(
-
x
+
0(x
+)
ie
.
x
= 1
-
x
+
x
+
..
.
+
ex
+
0xn
t
,
If)
InC1
x
=
x
-
+
+
..
+
74
*
+
0x
4)
x5
19
1
ta
n
"
(
x
)
=
x
-
=
+
5
-
...
+
+
0x
4
(-
1)
(h)
(1
+
x)
*
=
H
xx
+
"x
4
...
+
a
-
+
x
+
0x
Ta
y
l
o
r
series
Pu(
X)
n
+
o
In
st
ead
of
or
de
r
a
po
l
y
n
o
mi
a
l
,
we
ar
e
us
ing
a
series
·
Theor
em
(Uniq
ueness)
Assume
the
ex
i
s
ta
n
c
e
of
Ta
y
l
o
r
po
lyno
mial
(o
r
Ta
y
l
o
r
ser
i
es]
ne
ar
x
=
a
then
Ta
y
l
o
r
po
lyno
mial
(o
r
Ta
y
l
o
r
Se
r
ie
s)
is
unique
.
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Ex
ample
Ev
aluat
e
2
Sinx
-
sin(2x)
lim
29
x
-
2
-
2x
-
x
"
X
+
0
A
ation
2
.
(x
-
+
01
x
)
-
12
x)
-
2
+
01
5
3
)
lim
X
+
O
21
1
+
x
+
+
+
01
x4
)
)
-
2
-
2x
-
x
x
+
54
x
+
0(x)
-
3
=lim
x
-
0
x
+
0(x
Y)
5
=
(5
+
2)
+
0I
x
Y
-
*
=
3
.
-
lim
I
5
+
O(
X)
5
x
+
0
ALL
RIGHTS
RESER
VED.
wanminliu@gmail.com
Ex
ample
Find
the
Ta
y
l
o
r
po
lyn
o
mial
of
ta
n
h
"
(
x
)
ne
a
r
x-
o
up
to
or
der
2n
+
1
.
Hint
ta
n
h
"
(
x
)
=
Ie
n
(
*
S
1
ta
n
h
"
(x)
=
I
In
1
*
(
=>
I(
(
n[
1
+
x)
-
(((
-
x))
x2
=
I
I
X
-
I
+
...
+
(1)2x
4
+
01
x
2
2n+
1
,
+x
x
2n
-
(x
-
I
.....
-
)
***
+
01
x
+
]
ev
e
n
te
r
m
s
va
n
i
s
h
I
0
+
2
.
X
2n
+
1
=
I
[2
.
x
+
0
+
2
..
+
-
+
0(x2
+]
4
2n+
1
odd
te
r
m
s
sur
viv
e
3
=
x
+
+
+
...
+
+
0xxn
+
,
1